1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground_2/relocation/rtmap_id.ma".
16 include "static_2/notation/relations/relation_4.ma".
17 include "static_2/syntax/cext2.ma".
18 include "static_2/relocation/sex.ma".
19 include "static_2/static/frees.ma".
21 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
23 definition rex (R) (T): relation lenv ≝
24 λL1,L2. ∃∃f. L1 ⊢ 𝐅+⦃T⦄ ≘ f & L1 ⪤[cext2 R,cfull,f] L2.
26 interpretation "generic extension on referred entries (local environment)"
27 'Relation R T L1 L2 = (rex R T L1 L2).
29 definition R_confluent2_rex: relation4 (relation3 lenv term term)
30 (relation3 lenv term term) … ≝
32 ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
33 ∀L1. L0 ⪤[RP1,T0] L1 → ∀L2. L0 ⪤[RP2,T0] L2 →
34 ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
36 definition rex_confluent: relation … ≝
38 ∀K1,K,V1. K1 ⪤[R1,V1] K → ∀V. R1 K1 V1 V →
39 ∀K2. K ⪤[R2,V] K2 → K ⪤[R2,V1] K2.
41 definition rex_transitive: relation3 ? (relation3 ?? term) … ≝
43 ∀K1,K,V1. K1 ⪤[R1,V1] K →
44 ∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2.
46 (* Basic inversion lemmas ***************************************************)
48 lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R,T] Y2 → Y2 = ⋆.
49 #R #Y2 #T * /2 width=4 by sex_inv_atom1/
52 lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R,T] ⋆ → Y1 = ⋆.
53 #R #I #Y1 * /2 width=4 by sex_inv_atom2/
56 lemma rex_inv_sort (R):
57 ∀Y1,Y2,s. Y1 ⪤[R,⋆s] Y2 →
59 | ∃∃I1,I2,L1,L2. L1 ⪤[R,⋆s] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
60 #R * [ | #Y1 #I1 ] #Y2 #s * #f #H1 #H2
61 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
62 | lapply (frees_inv_sort … H1) -H1 #Hf
63 elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
64 elim (sex_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
65 /5 width=7 by frees_sort, ex3_4_intro, ex2_intro, or_intror/
69 lemma rex_inv_zero (R):
70 ∀Y1,Y2. Y1 ⪤[R,#0] Y2 →
72 | ∃∃I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 & R L1 V1 V2 &
73 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
74 | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R,cfull,f] L2 &
75 Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
76 #R * [ | #Y1 * #I1 [ | #X ] ] #Y2 * #f #H1 #H2
77 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/
78 | elim (frees_inv_unit … H1) -H1 #g #HX #H destruct
79 elim (sex_inv_next1 … H2) -H2 #I2 #L2 #HL12 #H #H2 destruct
80 >(ext2_inv_unit_sn … H) -H /3 width=8 by or3_intro2, ex4_4_intro/
81 | elim (frees_inv_pair … H1) -H1 #g #Hg #H destruct
82 elim (sex_inv_next1 … H2) -H2 #Z2 #L2 #HL12 #H
83 elim (ext2_inv_pair_sn … H) -H
84 /4 width=9 by or3_intro1, ex4_5_intro, ex2_intro/
88 lemma rex_inv_lref (R):
89 ∀Y1,Y2,i. Y1 ⪤[R,#↑i] Y2 →
91 | ∃∃I1,I2,L1,L2. L1 ⪤[R,#i] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
92 #R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2
93 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
94 | elim (frees_inv_lref … H1) -H1 #g #Hg #H destruct
95 elim (sex_inv_push1 … H2) -H2
96 /4 width=7 by ex3_4_intro, ex2_intro, or_intror/
100 lemma rex_inv_gref (R):
101 ∀Y1,Y2,l. Y1 ⪤[R,§l] Y2 →
102 ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
103 | ∃∃I1,I2,L1,L2. L1 ⪤[R,§l] L2 & Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
104 #R * [ | #Y1 #I1 ] #Y2 #l * #f #H1 #H2
105 [ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
106 | lapply (frees_inv_gref … H1) -H1 #Hf
107 elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
108 elim (sex_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
109 /5 width=7 by frees_gref, ex3_4_intro, ex2_intro, or_intror/
113 (* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *)
114 lemma rex_inv_bind (R):
115 ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
116 ∧∧ L1 ⪤[R,V1] L2 & L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2.
117 #R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
118 /6 width=6 by sle_sex_trans, sex_inv_tl, ext2_pair, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
121 (* Basic_2A1: uses: llpx_sn_inv_flat *)
122 lemma rex_inv_flat (R):
123 ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 →
124 ∧∧ L1 ⪤[R,V] L2 & L1 ⪤[R,T] L2.
125 #R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
126 /5 width=6 by sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
129 (* Advanced inversion lemmas ************************************************)
131 lemma rex_inv_sort_bind_sn (R):
132 ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R,⋆s] L2 →
133 ∃∃I2,K2. K1 ⪤[R,⋆s] K2 & L2 = K2.ⓘ{I2}.
134 #R #I1 #K1 #L2 #s #H elim (rex_inv_sort … H) -H *
136 | #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
140 lemma rex_inv_sort_bind_dx (R):
141 ∀I2,K2,L1,s. L1 ⪤[R,⋆s] K2.ⓘ{I2} →
142 ∃∃I1,K1. K1 ⪤[R,⋆s] K2 & L1 = K1.ⓘ{I1}.
143 #R #I2 #K2 #L1 #s #H elim (rex_inv_sort … H) -H *
145 | #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
149 lemma rex_inv_zero_pair_sn (R):
150 ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R,#0] L2 →
151 ∃∃K2,V2. K1 ⪤[R,V1] K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
152 #R #I #L2 #K1 #V1 #H elim (rex_inv_zero … H) -H *
154 | #Z #Y1 #K2 #X1 #V2 #HK12 #HV12 #H1 #H2 destruct
155 /2 width=5 by ex3_2_intro/
156 | #f #Z #Y1 #Y2 #_ #_ #H destruct
160 lemma rex_inv_zero_pair_dx (R):
161 ∀I,L1,K2,V2. L1 ⪤[R,#0] K2.ⓑ{I}V2 →
162 ∃∃K1,V1. K1 ⪤[R,V1] K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
163 #R #I #L1 #K2 #V2 #H elim (rex_inv_zero … H) -H *
165 | #Z #K1 #Y2 #V1 #X2 #HK12 #HV12 #H1 #H2 destruct
166 /2 width=5 by ex3_2_intro/
167 | #f #Z #Y1 #Y2 #_ #_ #_ #H destruct
171 lemma rex_inv_zero_unit_sn (R):
172 ∀I,K1,L2. K1.ⓤ{I} ⪤[R,#0] L2 →
173 ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 & L2 = K2.ⓤ{I}.
174 #R #I #K1 #L2 #H elim (rex_inv_zero … H) -H *
176 | #Z #Y1 #Y2 #X1 #X2 #_ #_ #H destruct
177 | #f #Z #Y1 #K2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
181 lemma rex_inv_zero_unit_dx (R):
182 ∀I,L1,K2. L1 ⪤[R,#0] K2.ⓤ{I} →
183 ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R,cfull,f] K2 & L1 = K1.ⓤ{I}.
184 #R #I #L1 #K2 #H elim (rex_inv_zero … H) -H *
186 | #Z #Y1 #Y2 #X1 #X2 #_ #_ #_ #H destruct
187 | #f #Z #K1 #Y2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
191 lemma rex_inv_lref_bind_sn (R):
192 ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R,#↑i] L2 →
193 ∃∃I2,K2. K1 ⪤[R,#i] K2 & L2 = K2.ⓘ{I2}.
194 #R #I1 #K1 #L2 #i #H elim (rex_inv_lref … H) -H *
196 | #Z1 #I2 #Y1 #K2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
200 lemma rex_inv_lref_bind_dx (R):
201 ∀I2,K2,L1,i. L1 ⪤[R,#↑i] K2.ⓘ{I2} →
202 ∃∃I1,K1. K1 ⪤[R,#i] K2 & L1 = K1.ⓘ{I1}.
203 #R #I2 #K2 #L1 #i #H elim (rex_inv_lref … H) -H *
205 | #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
209 lemma rex_inv_gref_bind_sn (R):
210 ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R,§l] L2 →
211 ∃∃I2,K2. K1 ⪤[R,§l] K2 & L2 = K2.ⓘ{I2}.
212 #R #I1 #K1 #L2 #l #H elim (rex_inv_gref … H) -H *
214 | #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
218 lemma rex_inv_gref_bind_dx (R):
219 ∀I2,K2,L1,l. L1 ⪤[R,§l] K2.ⓘ{I2} →
220 ∃∃I1,K1. K1 ⪤[R,§l] K2 & L1 = K1.ⓘ{I1}.
221 #R #I2 #K2 #L1 #l #H elim (rex_inv_gref … H) -H *
223 | #I1 #Z2 #K1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
227 (* Basic forward lemmas *****************************************************)
229 lemma rex_fwd_zero_pair (R):
230 ∀I,K1,K2,V1,V2. K1.ⓑ{I}V1 ⪤[R,#0] K2.ⓑ{I}V2 → K1 ⪤[R,V1] K2.
231 #R #I #K1 #K2 #V1 #V2 #H
232 elim (rex_inv_zero_pair_sn … H) -H #Y #X #HK12 #_ #H destruct //
235 (* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
236 lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R,②{I}V.T] L2 → L1 ⪤[R,V] L2.
237 #R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
238 [ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf
239 /4 width=6 by sle_sex_trans, sor_inv_sle_sn, ex2_intro/
242 (* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *)
243 lemma rex_fwd_bind_dx (R):
244 ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R,ⓑ{p,I}V1.T] L2 →
245 R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,T] L2.ⓑ{I}V2.
246 #R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (rex_inv_bind … H HV) -H -HV //
249 (* Basic_2A1: uses: llpx_sn_fwd_flat_dx *)
250 lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R,ⓕ{I}V.T] L2 → L1 ⪤[R,T] L2.
251 #R #I #L1 #L2 #V #T #H elim (rex_inv_flat … H) -H //
254 lemma rex_fwd_dx (R):
255 ∀I2,L1,K2,T. L1 ⪤[R,T] K2.ⓘ{I2} →
256 ∃∃I1,K1. L1 = K1.ⓘ{I1}.
257 #R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
258 [ elim (sex_inv_push2 … Hf) | elim (sex_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct
259 /2 width=3 by ex1_2_intro/
262 (* Basic properties *********************************************************)
264 lemma rex_atom (R): ∀I. ⋆ ⪤[R,⓪{I}] ⋆.
265 #R * /3 width=3 by frees_sort, frees_atom, frees_gref, sex_atom, ex2_intro/
269 ∀I1,I2,L1,L2,s. L1 ⪤[R,⋆s] L2 → L1.ⓘ{I1} ⪤[R,⋆s] L2.ⓘ{I2}.
270 #R #I1 #I2 #L1 #L2 #s * #f #Hf #H12
271 lapply (frees_inv_sort … Hf) -Hf
272 /4 width=3 by frees_sort, sex_push, isid_push, ex2_intro/
276 ∀I,L1,L2,V1,V2. L1 ⪤[R,V1] L2 →
277 R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R,#0] L2.ⓑ{I}V2.
278 #R #I1 #I2 #L1 #L2 #V1 *
279 /4 width=3 by ext2_pair, frees_pair, sex_next, ex2_intro/
283 ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R,cfull,f] L2 →
284 L1.ⓤ{I} ⪤[R,#0] L2.ⓤ{I}.
285 /4 width=3 by frees_unit, sex_next, ext2_unit, ex2_intro/ qed.
288 ∀I1,I2,L1,L2,i. L1 ⪤[R,#i] L2 → L1.ⓘ{I1} ⪤[R,#↑i] L2.ⓘ{I2}.
289 #R #I1 #I2 #L1 #L2 #i * /3 width=3 by sex_push, frees_lref, ex2_intro/
293 ∀I1,I2,L1,L2,l. L1 ⪤[R,§l] L2 → L1.ⓘ{I1} ⪤[R,§l] L2.ⓘ{I2}.
294 #R #I1 #I2 #L1 #L2 #l * #f #Hf #H12
295 lapply (frees_inv_gref … Hf) -Hf
296 /4 width=3 by frees_gref, sex_push, isid_push, ex2_intro/
299 lemma rex_bind_repl_dx (R):
300 ∀I,I1,L1,L2,T. L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I1} →
301 ∀I2. cext2 R L1 I I2 → L1.ⓘ{I} ⪤[R,T] L2.ⓘ{I2}.
302 #R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR
303 /3 width=5 by sex_pair_repl, ex2_intro/
306 (* Basic_2A1: uses: llpx_sn_co *)
307 lemma rex_co (R1) (R2):
308 (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
309 ∀L1,L2,T. L1 ⪤[R1,T] L2 → L1 ⪤[R2,T] L2.
310 #R1 #R2 #HR #L1 #L2 #T * /5 width=7 by sex_co, cext2_co, ex2_intro/
313 lemma rex_isid (R1) (R2):
315 (∀f. L1 ⊢ 𝐅+⦃T1⦄ ≘ f → 𝐈⦃f⦄) →
316 (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅+⦃T2⦄ ≘ f) →
317 L1 ⪤[R1,T1] L2 → L1 ⪤[R2,T2] L2.
318 #R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
319 /4 width=7 by sex_co_isid, ex2_intro/
322 lemma rex_unit_sn (R1) (R2):
323 ∀I,K1,L2. K1.ⓤ{I} ⪤[R1,#0] L2 → K1.ⓤ{I} ⪤[R2,#0] L2.
324 #R1 #R2 #I #K1 #L2 #H
325 elim (rex_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct
326 /3 width=7 by rex_unit, sex_co_isid/
329 (* Basic_2A1: removed theorems 9:
330 llpx_sn_skip llpx_sn_lref llpx_sn_free
332 llpx_sn_Y llpx_sn_ge_up llpx_sn_ge
333 llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx